Abstract
We propose and analyze an unfitted finite element method of arbitrary order for solving elliptic problems on domains with curved boundaries and interfaces. The approximation space on the whole domain is obtained by the direct extension of the finite element space defined on interior elements, in the sense that there is no degree of freedom locating in boundary/interface elements. We apply a non-symmetric bilinear form and the boundary/jump conditions are imposed in a weak sense in the scheme. The method is shown to be stable without any mesh adjustment or any special stabilization. The optimal convergence rate under the energy norm is derived, and \(O(h^{-2})\)-upper bounds of the condition numbers are shown for the final linear systems. Numerical results in both two and three dimensions are presented to illustrate the accuracy and the robustness of the method.
Similar content being viewed by others
Data Availability
Enquiries about data availability should be directed to the authors.
References
Adams, R.A., Fournier, J.J.F.: Sobolev Spaces, Pure and Applied Mathematics (Amsterdam), vol. 140, 2nd edn. Elsevier/Academic Press, Amsterdam (2003)
Areias, P.M.A., Belytschko, T.: Letter to the editor: A comment on the article: “A finite element method for the simulation of strong and weak discontinuities in solid mechanics” [Comput. Methods Appl. Mech. Engrg. 193 (2004)(33-35), 3523–3540; mr2075053] by Hansbo, A., Hansbo, P. Comput. Methods Appl. Mech. Engrg. 195 (2006)(9-12), 1275–1276
Ayuso de Dios, B., Brezzi, F., Havle, O., Marini, L.D.: \(L^2\)-estimates for the DG IIPG-0 scheme. Numer. Methods Partial Diff. Equ. 28(5), 1440–1465 (2012)
Babuška, I., Banerjee, U.: Stable generalized finite element method (SGFEM). Comput. Methods Appl. Mech. Eng. 201(204), 91–111 (2012)
Badia, S., Verdugo, F., Martín, A.: The aggregated unfitted finite element method for elliptic problems. Comput. Methods Appl. Mech. Eng. 336, 533–553 (2018)
Belytschko, T., Gracie, R., Ventura, G.: A review of extended/generalized finite element methods for material modeling. Modelling Simul. Mater. Sci. Eng. 17(4), 043001 (2009)
Bordas, S.P.A., Burman, E., Larson, M.G., Olshanskii, M.A. (eds.), Geometrically unfitted finite element methods and applications. In: Lecture Notes in Computational Science and Engineering. Springer, Cham, 2017, Held (2016)
Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods, Texts in Applied Mathematics, vol. 15, 3rd edn. Springer, New York (2008)
Burman, E.: Ghost penalty. C. R. Math. Acad. Sci. Paris 348(21–22), 1217–1220 (2010)
Burman, E., Cicuttin, M., Delay, G., Ern, A.: An unfitted hybrid high-order method with cell agglomeration for elliptic interface problems. SIAM J. Sci. Comput. 43(2), A859–A882 (2021)
Burman, E., Claus, S., Hansbo, P., Larson, M.G., Massing, A.: CutFEM: discretizing geometry and partial differential equations. Int. J. Numer. Methods Eng. 104(7), 472–501 (2015)
Burman, E., Hansbo, P.: Fictitious domain finite element methods using cut elements: II. A stabilized Nitsche method. Appl. Numer. Math. 62(4), 328–341 (2012)
Burman, E., Hansbo, P.: Fictitious domain methods using cut elements: III. A stabilized Nitsche method for Stokes’ problem. ESAIM Math. Model. Numer. Anal. 48(3), 859–874 (2014)
Burman, E., Hansbo, P., Larson, M.G., Massing, A.: A stable cut finite element method for partial differential equations on surfaces: the Helmholtz-Beltrami operator. Comput. Methods Appl. Mech. Eng. 362, 112803, 21 (2020)
Cui, T., Leng, W., Liu, H., Zhang, L., Zheng, W.: High-order numerical quadratures in a tetrahedron with an implicitly defined curved interface. ACM Trans. Math. Softw. (2019), 46(1), 1–18 (2019)
de Prenter, F., Lehrenfeld, C., Massing, A.: A note on the stability parameter in Nitsche’s method for unfitted boundary value problems. Comput. Math. Appl. 75(12), 4322–4336 (2018)
Dolejší, V., Havle, O.: The \(L^2\)-optimality of the IIPG method for odd degrees of polynomial approximation in 1D. J. Sci. Comput. 42(1), 122–143 (2010)
Fries, T.-P., Belytschko, T.: The extended/generalized finite element method: an overview of the method and its applications. Int. J. Numer. Methods Eng. 84(3), 253–304 (2010)
Girault, V., Raviart, P.-A.: Finite element methods for Navier-Stokes equations. In: Springer-Verlag, Berlin, 1986
Gürkan, C., Massing, A.: A stabilized cut discontinuous Galerkin framework for elliptic boundary value and interface problems. Comput. Methods Appl. Mech. Eng. 348, 466–499 (2019)
Gürkan, C., Massing, A.: A stabilized cut discontinuous Galerkin framework for elliptic boundary value and interface problems. Comput. Methods Appl. Mech. Eng. 348, 466–499 (2019)
Gürkan, C., Sticko, S., Massing, A.: Stabilized cut discontinuous Galerkin methods for advection-reaction problems. SIAM J. Sci. Comput. 42(5), A2620–A2654 (2020)
Guzmán, J., Olshanskii, M.: Inf-sup stability of geometrically unfitted Stokes finite elements. Math. Comp. 87(313), 2091–2112 (2018)
Guzmán, J., Rivière, B.: Sub-optimal convergence of non-symmetric discontinuous Galerkin methods for odd polynomial approximations. J. Sci. Comput. 40(1–3), 273–280 (2009)
Han, Y., Chen, H., Wang, X., **e, X.: EXtended HDG methods for second order elliptic interface problems. J. Sci. Comput. 84(1), 29 (2020)
Hansbo, A., Hansbo, P.: An unfitted finite element method, based on Nitsche’s method, for elliptic interface problems. Comput. Methods Appl. Mech. Eng. 191(47–48), 5537–5552 (2002)
Hansbo, P., Larson, M.G.: Discontinuous Galerkin methods for incompressible and nearly incompressible elasticity by Nitsche’s method. Comput. Methods Appl. Mech. Eng. 191(17–18), 1895–1908 (2002)
Hansbo, P., Larson, M.G., Zahedi, S.: A cut finite element method for a Stokes interface problem. Appl. Numer. Math. 85, 90–114 (2014)
Huang, P., Wu, H., **ao, Y.: An unfitted interface penalty finite element method for elliptic interface problems. Comput. Methods Appl. Mech. Eng. 323, 439–460 (2017)
Johansson, A., Larson, M.G.: A high order discontinuous Galerkin Nitsche method for elliptic problems with fictitious boundary. Numer. Math. 123(4), 607–628 (2013)
Kellogg, R.B.: Higher order singularities for interface problems, The mathematical foundations of the finite element method with applications to partial differential equations. In: Proc. Sympos., Univ. Maryland, Baltimore, Md., pp. 589–602 (1972). MR 0433926
Kellogg, R.B., Osborn, J.E.: A regularity result for the Stokes problem in a convex polygon. J. Funct. Anal. 21(4), 397–431 (1976)
Kramer, R., Bochev, P., Siefert, C., Voth, T.: An extended finite element method with algebraic constraints (XFEM-AC) for problems with weak discontinuities. Comput. Methods Appl. Mech. Eng. 266, 70–80 (2013)
Larson, M.G., Niklasson, A.J.: Analysis of a family of discontinuous Galerkin methods for elliptic problems: the one dimensional case. Numer. Math. 99(1), 113–130 (2004)
Lehrenfeld, C.: High order unfitted finite element methods on level set domains using isoparametric map**s. Comput. Methods Appl. Mech. Eng. 300, 716–733 (2016)
Lehrenfeld, C., Reusken, A.: Analysis of a high-order unfitted finite element method for elliptic interface problems. IMA J. Numer. Anal. 38(3), 1351–1387 (2018)
Li, K., Atallah, N.-M., Main, G.-A., Scovazzi, G.: The shifted interface method: a flexible approach to embedded interface computations. Int. J. Numer. Methods Eng. 121(3), 492–518 (2020)
Li, R., Yang, F.: A discontinuous Galerkin method by patch reconstruction for elliptic interface problem on unfitted mesh. SIAM J. Sci. Comput. 42(2), A1428–A1457 (2020)
Li, Z.: The immersed interface method using a finite element formulation. Appl. Numer. Math. 27(3), 253–267 (1998)
Lin, T., Lin, Y., Zhang, X.: Partially penalized immersed finite element methods for elliptic interface problems. SIAM J. Numer. Anal. 53(2), 1121–1144 (2015)
Main, A., Scovazzi, G.: The shifted boundary method for embedded domain computations. Part I: poisson and Stokes problems. J. Comput. Phys. 372, 972–995 (2018)
Massing, A., Larson, M.G., Logg, A., Rognes, M.E.: A stabilized Nitsche fictitious domain method for the Stokes problem. J. Sci. Comput. 61(3), 604–628 (2014)
Neiva, E., Badia, S.: Robust and scalable \(h\)-adaptive aggregated unfitted finite elements for interface elliptic problems. Comput. Methods Appl. Mech. Eng. 380, 26 (2021)
Saye, R.I.: High-order quadrature methods for implicitly defined surfaces and volumes in hyperrectangles. SIAM J. Sci. Comput. 37(2), A993–A1019 (2015)
Strouboulis, T., Babuška, I., Copps, K.: The design and analysis of the generalized finite element method. Comput. Methods Appl. Mech. Eng. 181(1–3), 43–69 (2000)
Wei, Z., Li, C., Zhao, S.: A spatially second order alternating direction implicit (ADI) method for solving three dimensional parabolic interface problems. Comput. Math. Appl. 75(6), 2173–2192 (2018)
Wu, H., **ao, Y.: An unfitted \(hp\)-interface penalty finite element method for elliptic interface problems. J. Comput. Math. 37(3), 316–339 (2019)
Zhou, Y.C., Wei, G.W.: On the fictitious-domain and interpolation formulations of the matched interface and boundary (MIB) method. J. Comput. Phys. 219(1), 228–246 (2006)
Acknowledgements
The authors would like to thank the anonymous referee sincerely for the constructive comments that improve the quality of this paper. This work was supported by National Natural Science Foundation of China (12201442, 12171340, 11971041) and National Key R &D Program of China (2020YFA0714000).
Funding
The authors declare that they have no funding.